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# Argument selection for T565
This is the builder's justification, not the independent critique or acceptance.
The governing instructions are archived at
`agents/runs/20261027T233828Z-build-skills/manifest.json`. Both the local and live
skills have SHA-256 `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The identical manifest is attached as `skill-manifest.json`.
The single quantity is the symmetric Morris integral with normalized angular
measure, labelled coordinates, and weight |1+exp(i theta)|^(beta k). The
parameters, in citation order, are beta, n, k. There is no factorial division,
complex phase, root choice, or parameter that selects a different quantity.
## Dyson exponent beta
The exact identities are `1`, `2`, `4`. Category: documented_application.
These are the circular orthogonal, unitary and symplectic ensembles. In the
symplectic case n counts distinct eigenvalues once, not their doubled complex
representation. The orthogonal and symplectic circular ensembles are symmetric
and selfdual unitary matrices; they are not Haar orthogonal and Haar symplectic
matrices. The same three values occur in the family proposal #225.
Checked source: Nishigaki, *Eigenphase distributions of unimodular circular
ensembles*, published equation (2), or arXiv:2401.09045v2 HTML equation (4).
The source first defines the unconstrained circular ensemble before imposing
its separate determinant constraint. Our integral is unconstrained.
## Number of remaining particles n
Category: discrete_index. Domain: every positive integer. Selection: every
integer from 1 through 10, for each chosen beta and k. It is an actual particle
count, rather than a continuous coupling renamed as an integer. These small
finite systems provide exact normalizations and low-dimensional local
correlation coefficients. Ten is a noncanonical practical reference cutoff.
The scalar n=1 boundary, even and odd Pfaffian dimensions, and three symmetry
classes are all covered before extending the dimension further.
Checked source: Forrester–Warnaar (1.17) defines the n-dimensional integral;
(4.12) gives the coalescence factor with n remaining and k coalescing particles
when the source's total particle count is replaced by n+k.
## Number of coalescing particles k
Category: mixed, with exhaustive disjoint groups:
* Identity `0`: intrinsic. The charge disappears and the Morris integral is
the Dyson normalization. Source: Forrester–Warnaar (1.18) and the sentence
immediately following it; Nishigaki's published (2).
* Identities `1`, `2`, `3`, `4`: documented_application. These count the
particles brought together in the local k-point correlation coefficient of
an ensemble of n+k particles, giving a=b=k beta/2 in the Morris integral.
Source: Forrester–Warnaar (4.12). The k=1 boundary also fixes one eigenangle
by rotational invariance. For beta=2 the charged integral divided by n! is
the characteristic-polynomial moment E|det(I+U)|^(2k) for Haar U(n).
The finite bound k<=4 is noncanonical. It includes the uncharged baseline,
single-particle insertion and the first several cluster orders across all
three ensembles without pursuing large moment orders solely to add entries.
The even and odd charge powers in the orthogonal case are both retained.
## Coverage and extent
The default generator enumerates beta in (1,2,4), then n=1,...,10, then
k=0,...,4, for 150 entries. This rectangular selection consists of three
symmetry classes and two genuine particle counts. It is not a product of
exploratory real or rational input grids. No group is a reference_sample, and
no reference_policy is needed for this selection. The finite bounds themselves
are computational and editorial choices, not intrinsic endpoints of the
mathematical domains. Breadth covers the three ensembles, uncharged and charged
regimes, and low-order coalescence before increasing either index.
Measured results support retaining the complete proposed extent: 103 exact
integers, 47 irrational values at 100 significant digits, longest serialized
value 101 characters, and a dry-run block estimate of 14.7 KiB. There is no
size exception. This does not imply every member is separately famous, nor
does cheap evaluation alone justify the family. The actual application and
normalization fix the arguments. Quantiles and statistical critical values
play no role.
## Sources checked
* https://people.smp.uq.edu.au/OleWarnaar/pubs/Selberg_review.pdf,
equations (1.17), (1.18), (4.12).
* https://academic.oup.com/ptep/article/2024/2/021A02/7602387,
equations (2), (6); browser text was readable, direct download returned 403.
The accessible author version is https://arxiv.org/html/2401.09045v2,
equations (4), (8) in that HTML numbering.
The coalescence application agrees exactly with the source and the batch.
No continuous charge extension or general asymmetric Morris weight is selected.